Abstract

Two complex matrix pairs (A, B) and (A′, B′) are contragrediently equivalent if there are nonsingular S and R such that (A′, B′) = (S−1AR, R−1BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + A͠, B + B͠) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of A͠ and B͠. Each perturbation (A͠, B͠) of (A, B) defines the first order induced perturbation AB͠ + A͠B of the matrix AB, which is the first order summand in the product (A + A͠)(B + B͠) = AB + AB͠ + A͠B + A͠B͠. We find all canonical matrix pairs (A, B), for which the first order induced perturbations AB͠ + A͠B are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations ẋ = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S−1CS and (A, B) by contragredient equivalence transformations (S−1AR, R−1BS).

Abstract

Two complex matrix pairs (A, B) and (A′, B′) are contragrediently equivalent if there are nonsingular S and R such that (A′, B′) = (S−1AR, R−1BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + A͠, B + B͠) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of A͠ and B͠. Each perturbation (A͠, B͠) of (A, B) defines the first order induced perturbation AB͠ + A͠B of the matrix AB, which is the first order summand in the product (A + A͠)(B + B͠) = AB + AB͠ + A͠B + A͠B͠. We find all canonical matrix pairs (A, B), for which the first order induced perturbations AB͠ + A͠B are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations ẋ = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S−1CS and (A, B) by contragredient equivalence transformations (S−1AR, R−1BS).

1 Introduction

We study a matrix differential equation ẋ = ABx, whose matrix is a product of an m × n complex matrix A and an n × m complex matrix B. It is equivalent to ẏ = S−1ARR−1BSy, in which S and R are nonsingular matrices and x = Sy. Thus, we can reduce (A, B) by transformations of contragredient equivalence

(A,B)↦(S−1AR,R−1BS),S and R are nonsingular.(1)

The canonical form of (A, B) with respect to these transformations was obtained by Dobrovol′skaya and Ponomarev [3] and, independently, by Horn and Merino [5]:

each pair(A,B)is contragrediently equivalent to a direct sum, uniquely determined up topermutation of summands, of pairs of the types(Ir,Jr(λ)),(Jr(0),Ir),(Fr,Gr),(Gr,Fr),(2)

in which r = 1, 2, …,

Jr(λ):=λ10λ⋱⋱10λ(λ∈C),Fr:=001⋱⋱001,Gr:=100⋱⋱001

are r × r, r × (r − 1), (r − 1) × r matrices, and

(A1,B1)⊕(A2,B2):=(A1⊕A2,B1⊕B2).

Note that (F1, G1) = (010, 010); we denote by 0mn the zero matrix of size m × n, where m, n ∈ {0, 1, 2, …}. All matrices that we consider are complex matrices. All matrix pairs that we consider are counter pairs: a matrix pair (A, B) is a counter pair if A and BT have the same size.

A notion of miniversal deformation was introduced by Arnold [1, 2]. He constructed a miniversal deformation of a Jordan matrix J; i.e., a simple normal form to which all matrices J + E close to J can be reduced by similarity transformations that smoothly depend on the entries of E. García-Planas and Sergeichuk [4] constructed a miniversal deformation of a canonical pair (2) for contragredient equivalence (1).

For a counter matrix pair (A, B), we consider all matrix pairs (A + A͠, B + B͠) that are sufficiently close to (A, B). The pair (A͠, B͠) is called a perturbation of (A, B). Each perturbation (A͠, B͠) of (A, B) defines the induced perturbationAB͠ + A͠B + A͠B͠ of the matrix AB that is obtained as follows:

(A+A~)(B+B~)=AB+AB~+A~B+A~B~.

Since A͠ and B͠ are small, their product A͠B͠ is “very small”; we ignore it and consider only first order induced perturbationsAB͠ + A͠B of AB.

In this paper, we describe all canonical matrix pairs (A, B) of the form (2), for which the first order induced perturbations AB͠ + A͠B are nonzero for all miniversal perturbations (A͠, B͠) ≠ 0 in the normal form defined in [4].

Note that z = ABx can be considered as the superposition of the systems y = Bx and z = Ay:

x⟶B→yA⟶zimpliesx⟶AB⟶z

2 Miniversal deformations of counter matrix pairs

In this section, we recall the miniversal deformations of canonical pairs (2) for contragredient equivalence constructed by García-Planas and Sergeichuk [4].

N and H are matrices of the form (5) and (6), and the stars denote independent parameters.

Theorem 1

(see [4]). Let (A, B) be the canonical pair(3). Then all matrix pairs (A + A͠, B + B͠) that are sufficiently close to (A, B) are simultaneously reduced by some transformation

(A+A~,B+B~)↦(S−1(A+A~)R,R−1(B+B~)S),

in whichSandRare matrix functions that depend holomorphically on the entries ofA͠andB͠, S(0) = I, and R(0) = I, to the form(4), whose stars are replaced by complex numbers that depend holomorphically on the entries ofA͠andB͠. The number of stars is minimal that can be achieved by such transformations.

3 Main theorem

Each matrix pair (A + A͠, B + B͠) of the form (4), in which the stars are complex numbers, we call a miniversal normal pair and (A͠, B͠) a miniversal perturbation of (A, B).

The following theorem is the main result of the paper.

Theorem 2

Let (A, B) be a canonical pair(2). The following two conditions are equivalent:

AB͠ + A͠B ≠ 0 for all nonzero miniversal perturbations (Ã, B̃).

(A, B) does not contain

(Ir, Jr(0)) ⊕ (Jr(0), Ir) for each r,

(F1, G1) ⊕ (G2, F2), and

(Fm, Gm) ⊕ (Gm, Fm) for each m.

Proof

(a) ⟹ (b). Let (A, B) be a canonical pair (2). We should prove that if (A, B) contains a pair of type (i), (ii), or (iii), then AB͠ + A͠B = 0 for some miniversal perturbation (A͠, B͠) ≠ (0, 0). It is sufficient to prove this statement for (A, B) of types (i)–(iii).

Case 1

(A, B) = (Ir, Jr(0)) ⊕ (Jr(0), Ir) for some r. We should prove that there exists a nonzero miniversal perturbation (A͠, B͠) such that AB͠ + A͠B = 0.

If r is arbitrary, then the miniversal deformation (4) of (A, B) has the form

10εr⋱⋮001ε2r−2ε1…εr−1ε2r−10100⋱⋱001,010⋱⋱0001100⋱01α1α2…αrαr+1…α2r−1

in which all αi and εj are independent parameters. Since the rth row of B is zero, a parameter ε2r−2 does not appear in A͠B, and so in AB͠ + A͠B too. Choosing all parameters zeros except for ε2r−2 ≠ 0, we get AB͠ + A͠B = 0.